### Gerd Faltings

Born 28 July 1954, Gelsenkirchen-Buer, Germany

Fields Medal (1986) “for proving the Mordell Conjecture, using methods of arithmetic algebraic geometry.”

Gerd Faltings comes from a natural sciences oriented household – his father is a physicist, his mother a chemist. Whilst at school, Faltings twice won the ‘Bundeswettbewerb Mathematik’, a national mathematics competition. He studied Mathematics and Physics (1972-78) and received his Ph.D. 1978 under Hans-Joachim Nastold at the University of Münster. After a research fellowship at Harvard University, USA, (1978-79), he completed his post-doctoral training at the University of Münster (1979-81) and was then appointed to a professorship at the University of Wuppertal (1982-1984). In 1983 Faltings published a short paper on ‘Endlichkeitssätze für Abelsche Varietäten über Zahlkörpern’, for which he would become famous. He held a professorship at Princeton University, USA, from 1985 until 1994 when he returned to Germany to work at the Max Planck Institute for Mathematics in Bonn. He has been Director of that Institute since 1995.

In addition to the Fields Medal, Gerd Faltings has also received numerous other awards, including the Leibniz Prize of the Deutsche Forschungsgemeinschaft (1996), the Karl Georg Christian von Staudt Prize (2008), the Heinz Gumin Prize (2010) and the Federal Cross of Merit 1st Class (2009). In 1988, Faltings was awarded a Guggenheim Fellowship.

Gerd Faltings is a widowed and has two daughters. He enjoys gardening and likes to go to the opera.

Probably no other mathematician has shaped algebraic geometry since 1970 as Gerd Faltings. He became famous with a short paper – just 17 pages long – in which he proved, among other things, a conjecture by the American mathematician Louis Joel Mordell dating from 1922. Mordell had investigated algebraic curves. Such curves are defined by polynomials in one or more variables, that is: by expressions such as x2 + xy + y3; the curves consist of all points where the polynomial is zero. (Everyone knows from school the algebraic curve x2 + y2 – 1, which describes a simple circle with radius 1 around the origin in the plane.)

To each algebraic curve, one can assign a well-defined number, its genus. The genus determines the type of the curve: curves of genus zero are related to straight lines and curves of genus one are called elliptic curves. Curves of higher genus are either hyper-elliptic or non-hyper-elliptic, and Mordell’s conjecture said that these curves with a genus greater than one can never have an infinite number of rational points, i.e. points whose coordinates can be expressed as fractions (something like 1/2 or 4/5). Faltings proved that this conjecture is indeed correct. The result proved to be not only extremely important for algebraic geometry, but was also significant in the search for a proof of Fermat’s Last Theorem. Faltings’ result implied that for each n>2 only a finite number of coprime integers x, y and z exist such that the equation xⁿ + yⁿ = zⁿ is satisfied; Fermat’s theorem states, however, that there are no such numbers. The proof of the Mordell conjecture made Faltings famous, but he went on to make many other important contributions in several areas of algebraic geometry and topology. He himself describes his fields of research as follows: Diophantine equations, Arakelov theory, Abelian varieties, moduli spaces of vector bundles, p-adic Hodge theory.